Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.

翻译：近些年来,通过低维矢量空间展示知识图来预测缺失的事实或发现错误的事实,取得了巨大的成功。然而,目前还不能很好地了解如何以原则性的方式嵌入本体知识,例如,一套(存在)规则。为了解决这一缺陷,我们在本文件中引入了一个基于康韦克斯区域的框架,该框架可以忠实地将本体知识纳入矢量空间嵌入。我们的技术贡献是双重的。首先,我们表明一些最受欢迎的现有嵌入方法无法模拟甚至非常简单的规则类型。第二,我们表明我们的框架可以精确地代表使用所谓的准链状生存规则表达的本体,因此,利用矢量空间嵌入的任何一组事实在逻辑上是一致的,在与输入的本体学上是推算封闭的。